hep-lat/ May 1996

Transcription

1 DESY CERN-TH/ MPI-PhT/96-38 Chiral symmetry and O(a) improvement in lattice QCD Martin Luscher a, Stefan Sint b, Rainer Sommer c and Peter Weisz b hep-lat/ May 1996 a b c Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D Hamburg, Germany Max-Planck-Institut fur Physik, Fohringer Ring 6, D Munchen, Germany CERN, Theory Division, CH-1211 Geneve 23, Switzerland Abstract The dominant cuto eects in lattice QCD with Wilson quarks are proportional to the lattice spacing a. In particular, the isovector axial current satises the PCAC relation only up to such eects. Following a suggestion of Symanzik, they can be cancelled by adding local O(a) correction terms to the action and the axial current. We here address a number of theoretical issues in connection with the O(a) improvement of lattice QCD and then show that chiral symmetry can be used to x the coecients multiplying the correction terms. May 1996

2 1. Introduction A well-known decit of Wilson's formulation of lattice QCD is that the chiral symmetry of the theory is not preserved [1]. This is usually not considered to be a fundamental problem, because there is every reason to expect that the symmetry is restored in the continuum limit. But one should bear in mind that numerical simulations of lattice QCD are limited to relatively large lattice spacings a. Chiral symmetry may then still be rather strongly violated by lattice eects. A clear demonstration of this has recently been given in ref. [2]. The tests reported there have been performed for the case of quenched QCD with two mass-degenerate light quarks. In the continuum limit of this theory the isovector axial current A a is expected to satisfy the PCAC A a = 2mP a ; (1:1) where P a denotes the associated axial density and m the quark mass. By considering matrix elements of eq. (1.1) between various low-energy states it was found that the lattice corrections to the relation are far from being negligible on the accessible lattices. With the current simulation algorithms one needs at least a factor 32 more computer time to be able to reduce the lattice spacing by a factor 2 at constant statistical errors and physical length scales. At the same time the lattice eects (which are of order a in the Wilson theory) are only reduced by 50%. In view of this unfavourable situation it is evidently desirable to develop better strategies to bring the cuto eects under control. A comparatively simple and theoretically attractive possiblity is to apply Symanzik's improvement programme [3,4], where the O(a) cuto eects in onshell quantities (particle energies, scattering amplitudes, normalized matrix elements of local composite elds between particle states, etc.) are cancelled by adding local O(a) counterterms to the lattice action and to the composite elds of interest [5{8]. In the so constructed \on-shell improved" theory the continuum limit is reached much faster (with a rate proportional to a 2 ), while the computational cost of the terms added remains small [9{12]. On lattices without boundaries only one counterterm, the Sheikholeslami- Wohlert term [6], needs to be included in the lattice action and not many more are required to improve the low-dimensional operators such as the axial current and density [2,8]. A technical diculty however is that the coecients 1

3 multiplying these counterterms are not a priori known. They are functions of the gauge coupling and should be chosen so that the O(a) cuto eects in on-shell quantities cancel. The main observation made in this paper is that the coecient c sw multiplying the Sheikholeslami-Wohlert term can be determined by studying the violations of the PCAC relation (1.1) on the lattice. Chiral symmetry restoration and O(a) improvement thus get tied up in an interesting way. The idea works out both in perturbation theory and non-perturbatively, but here we only set up the theoretical framework and defer all detailed computations to two separate papers [23,24]. A short account of the method has already appeared in ref. [2]. The present paper also gives us the opportunity to discuss the problem of on-shell O(a) improvement from a somewhat novel point of view (sects. 2 and 3). In particular, the r^ole played by the O(a) counterterms proportional to the quark mass is claried, and it is shown that mass-independent renormalization schemes must be set up with care if O(a) improvement is to be preserved. The matrix elements used to study chiral symmetry restoration are constructed from the QCD Schrodinger functional [13{15,17] which we introduce in sect. 4. For completeness we also derive the form of the O(a) boundary counterterms that must be included in the action to improve the Schrodinger functional (sect. 5), although this is not really needed when we discuss the lattice corrections to the PCAC relation (1.1) in sect. 6. The paper ends with a few concluding remarks and a series of technical appendices. 2. On-shell O(a) improvement revisited Our aim in this section is to derive the form of the O(a) counterterms to the lattice action and the axial current and density that are required for on-shell improvement. Although the improved action has previously been obtained by Sheikholeslami and Wohlert [6], we think it is worthwhile to go through the argumentation again in a slightly dierent way. The extension of the discussion to the Schrodinger functional will then be rather easy (sect. 5). 2

4 2.1 Preliminaries In this section we choose to set up the theory on a four-dimensional hyper-cubic euclidean lattice with spacing a and innite extent in all directions. Most of our notational conventions are collected in appendix A. The gauge group is taken to be SU(N) and we assume for simplicity that there are N f avours of mass-degenerate light quarks, although it would not be very much more dicult to treat the case of light quarks with dierent masses. A gauge eld U on the lattice is an assignment of a matrix U(x; ) 2 SU(N) to every lattice point x and direction = 0; 1; 2; 3. Quark and antiquark elds, (x) and (x), reside on the lattice sites and carry Dirac, colour and avour indices. The (unimproved) lattice action is of the form S[U; ; ] = S G [U] + S F [U; ; ]; (2:1) where S G denotes the usual Wilson plaquette action and S F the Wilson quark action. Explicitly we have S G [U] = 1 X tr f1 g0 2 U(p)g (2:2) p with g 0 being the bare gauge coupling and U(p) the parallel transporter around the plaquette p. The sum runs over all oriented plaquettes p on the lattice. To dene the quark action S F we rst introduce the Wilson-Dirac operator D = 1 2 f (r + r ) ar r g ; (2:3) which involves the gauge covariant lattice derivatives r and r dened in appendix A. The action then assumes the standard form, S F [U; ; ] = a 4 X x (x)(d + m 0 ) (x); (2:4) where m 0 denotes the bare quark mass. 3

5 2.2 Local eective theory Close to the continuum limit the lattice theory dened above may be described in terms of a local eective theory with action S e = S 0 + as 1 + a 2 S 2 + : : : (2:5) The leading term, S 0, is just the action of the continuum theory, while the other terms are to be interpreted as operator insertions in the continuum theory. In his analysis of the cuto dependence of lattice eld theories, Symanzik [3,4] denes the continuum theory using dimensional regularization, but we could also employ a lattice with spacing very much smaller than a to give a precise meaning to S 0 and the operator insertions. The correction terms in the eective action are of the form S k = Z d 4 x L k (x); (2:6) where the lagrangians L k (x) are linear combinations of local composite elds of dimension 4 + k. The dimension counting here includes the (non-negative) powers of the quark mass m by which some of the elds may be multiplied. From the list of all possible such elds only a small subset occurs in the eective action. First of all, since one integrates over the position x, one can use partial integration to reduce the number of terms that must be included. The remaining terms must be invariant under gauge transformations and U(1)SU(N f ) avour rotations. They should also respect the exact discrete symmetries of the lattice theory. This includes all space-time lattice symmetries and the charge conjugation symmetry (appendix B). Taking these remarks into account one nds that the order a eective lagrangian, L 1 (x), must be a linear combination of the elds O 1 = F ; (2:7) O 2 = D D + D D ; (2:8) O 3 = m tr ff F g ; (2:9) O 4 = m D D ; (2:10) O 5 = m 2 ; (2:11) 4

6 where F denotes the gauge eld strength and D the gauge covariant partial derivative (cf. appendix A). In subsects. 2.3 and 2.4 we shall come back to this list of elds and show how it can be reduced to essentially one term. Cuto eects originate not only from the lattice action but also from the local composite elds that one is interested in. So let us consider some local gauge invariant eld (x) constructed from the quark and gluon elds on the lattice. For simplicity we assume that (x) does not mix with other elds under renormalization. We then expect that the connected renormalized n-point correlation functions G n (x 1 ; : : :; x n ) = (Z ) n h(x 1 ) : : :(x n )i con (2:12) have a well-dened continuum limit, provided the renormalization factor Z is chosen appropriately and if all points x 1 ; : : :; x n are kept at non-zero (physical) distances from one another. In the local eective theory the renormalized lattice eld Z (x) is represented by an eective eld e (x) = 0 (x) + a 1 (x) + a 2 2 (x) + : : : (2:13) The elds k (x) appearing here are linear combinations of local elds with the appropriate dimension and symmetry properties. To order a the lattice correlation functions are then given by G n (x 1 ; : : :; x n ) = h 0 (x 1 ) : : : 0 (x n )i con Z a d 4 y h 0 (x 1 ) : : : 0 (x n )L 1 (y)i con + a nx k=1 h 0 (x 1 ) : : : 1 (x k ) : : : 0 (x n )i con + O(a 2 ); (2:14) where the expectation values on the right hand side are to be taken in the continuum theory with action S 0. The second term is the contribution of the order a term in the eective action. Note that the integral over y in general diverges at the points y = x k. A subtraction prescription must hence be supplied. The precise way in which this happens is unimportant, because the arbitrariness that one has amounts to a local operator insertion at these points, i.e. to a redenition of the eld 1 (x). 5

7 It should also be emphasized that not all the dependence on the lattice spacing comes from the explicit factors of a in eq. (2.14). The other source of a-dependence are the elds 1 (x) and L 1 (y), which are linear combinations of some basis of elds. While the basis elements are independent of a, the coecients need not be so, although they are expected to vary only slowly with a. In perturbation theory the coecients are calculable polynomials in ln(a). 2.3 Cuto dependence of on-shell amplitudes Eventually all on-shell quantities in QCD can be extracted from correlation functions of local composite elds. An important point to note is that for this the correlation functions are only required at non-zero physical distances. We would now like to show that the local eective theory can be simplied considerably if attention is restricted to such correlation functions. As an example we discuss the functions G n introduced above, but the argumentation is readily carried over to correlation functions involving several kinds of elds. Let us rst consider the second term on the right hand side of eq. (2.14). The eective lagrangian L 1 (y) is a linear combination of the elds (2.7){(2.11). As long as y keeps away from the points x k, the eld equations of the continuum theory may be applied and one concludes that certain linear combinations of these elds do not contribute to the correlation function. This remains true after integration over y up to contact terms that arise at the points x 1 ; : : :; x k. Any such contact term amounts to an operator insertion, which, in eq. (2.14), can be compensated by a redenition of the eld 1 (the dimensionalities and symmetries of the elds involved do not allow for any other form of the contact terms). Since we do not insist on any particular expression for 1 (x), but only require that eq. (2.14) holds, we are hence free to apply the eld equations to simplify the eective lagrangian. The linear combinations of the elds (2.7){(2.11) that vanish are easily found at tree-level of perturbation theory by applying the classical eld equations. There are two such relations which allow us to eliminate O 2 and O 4. At non-zero couplings the coecients in the vanishing linear combinations change, but since we are only interested in reducing the number of basis elds, it is enough to know that the linear relations allow one to express some of them through the other basis elements. This information is invariant under small changes of the coecients so that, barring singular events, we may take L 1 to be a linear combination of O 1, O 3 and O 5. Similar arguments may be used to eliminate some of the terms that con- 6

8 x ν x µ Fig. 1. Graphical representation of the products of gauge eld variables contributing to the lattice eld strength tensor (2.18). Each square corresponds to one of the terms in eq. (2.19). tribute to the eld 1 (x). The only place in eq. (2.14) where 1 (x) appears is the third term on the right hand side. Since all positions x 1 ; : : :; x n are kept at non-zero distances from each other, no contact terms can arise when the eld equations are applied in this correlation function. The number of basis elds from which 1 (x) is constructed can thus be reduced as in the case of the eective lagrangian. 2.4 Improved lattice action Our aim is to construct an improved lattice action by adding a suitable O(a) counterterm to the Wilson action (2.1). The counterterm should be chosen such that the order a term in the eective action is cancelled. Since we are only interested in on-shell amplitudes, we may assume that the eective lagrangian L 1 is a linear combination of the elds O 1, O 3 and O 5. It is then quite obvious that L 1 can be made to vanish by adding a counterterm of the form a 5 X x n c 1 b O1 (x) + c 3 b O3 (x) + c 5 b O5 (x) o ; (2:15) where Ok b is some lattice representation of the eld O k. Apart from renormalizations of the bare parameters and adjustments of the coecients c k, the discretization ambiguities that one has here are of order a 2. In particular, we may choose to represent the elds tr ff F g and the Wilson plaquette eld and the local scalar density that already appears in the Wilson quark action. The O(a) counterterms proportional b O3 and b O5 then amount to a renormalization of the bare coupling and mass. At rst sight one 7 by

9 might think that such reparametrizations are insignicant, but as explained in sect. 3, this is not quite true if a mass-independent renormalization scheme is employed. For the time being we ignore this complication and simply drop the contributions of b O3 and b O5. For the on-shell O(a) improved action we thus obtain S impr [U; ; ] = S[U; ; ] + S[U; ; ]; (2:16) S[U; ; ] = a 5 X x c sw (x) i 4 b F (x) (x); (2:17) where S[U; ; ] is the Wilson action and b F a lattice representation of the gluon eld strength tensor F. We adopt the standard denition bf (x) = 1 8a 2 fq (x) Q (x)g ; (2:18) Q (x) = U(x; )U(x + a^; )U(x + a^; ) 1 U(x; ) 1 + U(x; )U(x a^ + a^; ) 1 U(x a^; ) 1 U(x a^; ) + U(x a^; ) 1 U(x a^ a^; ) 1 U(x a^ a^; )U(x a^; ) + U(x a^; ) 1 U(x a^; )U(x + a^ a^; )U(x; ) 1 : (2:19) The four terms in this equation correspond to the four plaquette loops shown in g. 1. The O(a) counterterm (2.17) has rst been written down by Sheikholeslami and Wohlert [6]. In their paper they perform eld transformations in the functional integral of the lattice theory to argue that only this term is required for on-shell improvement. Our strategy here has been to achieve a reduction of the number of terms already at the level of the eective action. This treatment is simpler in our opinion, because the discussion of the form of the O(a) correction terms takes place entirely in the continuum theory. The coecient c sw multiplying the Sheikholeslami-Wohlert term in the improved action is a function of the bare coupling g 0 and must be chosen so that the O(a) cuto eects in on-shell quantities cancel. It has been shown in ref. [6] that c sw = 1 to lowest order of perturbation theory. The coecient has 8

10 later been worked out to one-loop order by Wohlert [7]. His results will be rederived in ref. [23] and a non-perturbative determination of c sw, in the quenched approximation and for gauge group SU(3), will be reported in ref. [24]. 2.5 Improved axial current and density From the discussion in subsect. 2.2 it is clear that not all O(a) eects can be removed from the correlation functions of local composite elds by employing an improved action. One also has to use improved elds to achieve this. They are constructed in very much the same way as the improved action. So let us assume that (x) is some given (unimproved) composite eld on the lattice. The isovector axial current and density, A a (x) = (x) a (x); (2:20) P a (x) = (x) a (x); (2:21) where a is a Pauli matrix acting on the avour indices of the quark eld, are examples of such elds. We then determine the general form of the order a term, 1 (x), in the expansion (2.13) of the associated eective eld. Since we are only interested in the improvement of on-shell matrix elements, the classical eld equations may be used to eliminate some of the basis elds contributing to 1 (x) (cf. subsect. 2.3). The on-shell O(a) improved lattice eld is then given by I (x) = (x) + a(x); (2:22) where (x) is obtained by taking the general linear combination of a lattice representation of the remaining basis elds. Taking into account the transformation behaviour of the axial current (2.20) under the lattice symmetries and charge conjugation, it is straightforward to show that a list of all possible basis elds is in this case given by (O 6 ) a = a D D a ; (2:23) (O 7 ) a = 1 2 a 5 D + D a ; (2:24) (O 8 ) a = m a : (2:25) The rst of these can be related to the other two by the eld equations and so may be dropped. The O(a) counterterm associated with (O 8 ) a amounts 9

11 to a renormalization of the axial current. Since we have not imposed any renormalization conditions so far, we may at this point just as well drop this term (the issue will be addressed again in sect. 3). We are then left with the second term in the list above and conclude that A a (x) = c A 1 2 (@ )P a (x): (2:26) In the case of the axial density a similar analysis shows that P a (x) = 0; (2:27) where we have again ignored a mass-dependent renormalization factor. The coecient c A depends on the gauge coupling and is to be chosen so as to achieve the desired improvement. In perturbation theory c A is of order g 2 0, because at tree-level the local axial current (2.20) is already on-shell improved (up to the mass-dependent renormalization factor mentioned above). This has previously been noted in ref. [8]. The coecient is worked out to one-loop order of perturbation theory in ref. [23] and it is also possible to compute it non-perturbatively, using numerical simulations [24]. 3. Mass-independent renormalization schemes In this section we address the problem of the proper parametrization of the improved theory and clarify the r^ole played by the order a terms appearing in the local eective theory that correspond to renormalizations of the bare parameters and elds. 3.1 Renormalization and O(a) improvement In the improved theory the renormalization conditions on the gauge coupling, the quark mass and the improved composite elds must be chosen with care. What we would like to achieve is that the correlation functions of the renormalized elds, at xed non-zero physical distances and xed renormalized coupling and mass, converge to the continuum limit with a rate proportional to a 2. Of course this is only possible if the coecients c sw, c A, etc. have been assigned their proper values (which we assume is the case). 10

12 An obvious possibility is to impose all renormalization conditions on a set of renormalized correlation functions dened at the same point (g 0 ; am 0 ) in the bare parameter space. The renormalized amplitudes in such a scheme are evidently inert against transformations of the bare parameters and rescalings of the bare elds. Without loss the corresponding O(a) counterterms may hence be dropped (as we did in sect. 2). This also means that the complete list of counterterms has been taken into account and that, therefore, no uncancelled O(a) corrections can arise. In other words, renormalization schemes of this type are automatically compatible with O(a) improvement. A disadvantage of this procedure however is that the renormalized coupling and elds implicitly depend on the quark mass. Mass-independent schemes, where one imposes the renormalization conditions at zero quark mass, are intrinsically simpler and certainly better suited to discuss the scale evolution of the renormalized parameters [22]. The problem then is that the massive theory must be related to the massless theory, a link which is usually established via the bare parameters. As a result reparametrizations of the bare theory can no longer be ignored if O(a) improvement is to be preserved. These remarks will become clearer below, where we consider two examples for illustration. We shall then set up the general mass-independent renormalization scheme respecting O(a) improvement. Specic schemes are discussed later in this section. 3.2 Naive mass-independent schemes In the plane of bare parameters a critical line m 0 = m c (g 0 ) (3:1) is expected to exist, where the physical quark mass vanishes. Our aim is to parametrize the theory around this line. It is then useful to introduce the subtracted mass m q = m 0 m c : (3:2) As an aside we remark that m c (g 0 ) depends on how precisely the physical quark mass is dened. Dierent denitions lead to values of m c (g 0 ) that dier by terms of order a 2. The issue will be addressed again in subsect. 6.6 and also in ref. [24]. In this section order a 2 corrections are considered negligible and a precise denition of the critical bare mass is then not required. It is common to assume that the renormalized coupling g R and the renormalized mass m R in a mass-independent scheme are related to the bare param- 11

13 eters through where is a normalization mass and g 2 R = g 2 0Z g (g 2 0; a); (3:3) m R = m q Z m (g 2 0; a); (3:4) Z(g 2 0; a) = 1 + Z (1) (a)g Z (2) (a)g : : : (3:5) for Z = Z g and Z = Z m. We now show that such schemes always lead to uncancelled O(a) corrections in some renormalized amplitudes. At g 0 = 0 the quarks decouple and their dynamics is described by the free Wilson quark action. According to eqs. (3.4) and (3.5) (and since m c = 0 at zero coupling) the renormalized quark mass m R is equal to the bare mass m 0 in this situation. We are thus supposed to take the continuum limit at xed m 0, but as is well-known this leads to uncancelled O(a) corrections in various places. A particularly obvious case is the \pole mass", m p = 1 a ln(1 + am 0) = m R 1 2 am2 R + : : :; (3:6) which is equal to the energy of a free quark with zero momentum. It is possible to correct for this decit by replacing eq. (3.4) through m R = m q am q + O(g 2 0 ): (3:7) The pole mass m p then coincides with m R up to terms of order a 2. One might hope to get away with a modication of eq. (3.4), as suggested above, but it turns out that eq. (3.3) cannot be valid either if O(a) corrections in renormalized amplitudes are to be avoided. The argumentation is more dicult in this case, because the problem shows up only at one-loop order of perturbation theory. A relatively simple quantity to consider is the running coupling g 2 introduced in ref. [18]. g 2 is obtained from the QCD Schrodinger functional by computing the response of the functional to a change of the boundary values of the gauge eld. We do not need to know any further details about the coupling here except that it is a well-dened function of the spatial extent L of the lattice, the lattice spacing a and the bare parameters g 0 and m 0. 12

14 The perturbation expansion g 2 = g c (1;0) + c (1;1) N f g : : : (3:8) has been worked out in ref. [18]. The O(a) counterterms required for the improvement of the Schrodinger functional have been taken into account in this calculation (cf. sect. 5). If we insert the denition (3.7) of the renormalized quark mass, the result assumes the form c (1;0) = 11N 24 2 ln(l=a) + k 1 + O(a 2 ); (3:9) c (1;1) = ln(l=a) + k 2 + k 3 am R + O(a 2 ); (3:10) where k 2 is a function of m R L and k 3 = 0:012000(2): (3:11) We can now express g 2 as a series in the renormalized coupling g 2 R by eliminating the bare coupling. The renormalization factor Z g (1) (a) should be chosen so as to cancel the logarithmic divergence, but since it may not depend on the quark mass, we cannot get rid of the term proportional to k 3. We thus end up with an uncancelled O(a) correction and so conclude that eq. (3.3) must be modied to be compatible with O(a) improvement. 3.3 Improved mass-independent schemes From the discussion in sect. 2 we now recall that a complete O(a) improvement of the theory in general requires a renormalization of the bare parameters by factors of the form 1 + b(g 2 0)am q. We thus introduce a modied bare coupling and bare quark mass through ~g 2 0 = g 2 0 (1 + b g am q ) ; (3:12) em q = m q (1 + b m am q ) : (3:13) The coecients b g and b m depend on g 2 0 and should be chosen so as to cancel any remaining cuto eects of order a. Note that the modied and ordinary bare coupling coincide along the critical line m 0 = m c. The general massindependent renormalization scheme, compatible with O(a) improvement, is 13

15 now given by g 2 R = ~g 2 0Z g (~g 2 0; a); (3:14) m R = em q Z m (~g 2 0; a); (3:15) where Z g and Z m have an expansion of the form (3.5) with g 0 replaced by ~g 0. It may be worthwhile at this point to discuss the signicance of the modi- ed bare parameters ~g 0 and em q a little further. Our aim is to parametrize the bare theory in such a way that the continuum limit can be reached coherently for all quark masses and without O(a) corrections. To approach the limit the bare parameters have to be scaled in a particular way. The basic observation is that the scaling required for g 0 necessarily depends on the quark mass (if O(a) corrections are to be avoided), while ~g 0 scales independently of the quark mass. A similar comment applies to the bare quark mass m q, which must be scaled by a mass-dependent factor. The modied bare mass em q, on the other hand, is scaled by a factor depending on a only. It should be clear from these remarks that the coecients b g and b m are well-determined and independent of the particular renormalization scheme chosen. So far we have been exclusively concerned with the parameter renormalization, but it is now straightforward to extend the discussion to the renormalization of multiplicatively renormalizable local elds. Suppose (x) is such a eld and let I (x) be the associated improved eld [eq. (2.22)]. Following the conventions adopted in subsect. 2.5, the O(a) counterterm which amounts to a renormalization of the eld by a factor of the form 1+b(g 2 0)am q is not included in I (x). It seems more natural to us to include this factor in the denition R (x) = Z (~g 2 0 ; a)(1 + b am q ) I (x) (3:16) of the renormalized eld. The coecient b plays a r^ole completely analogous to b g and b m. In particular, it is independent of the renormalization condition chosen to x Z. In perturbation theory the b{coecients can be expanded according to b = b (0) + b (1) g b (2) g : : : (3:17) From the discussion in subsect. 3.2 we infer that b (0) g = 0 and b (1) g = 0:012000(2) N f : (3:18) 14

16 This suggests that the modied bare coupling ~g 2 0 is very nearly equal to g 2 0, for all couplings of interest and quark masses am q less than say 0:1. In practice the dierence can probably be ignored until very precise calculations become feasible. It should be noted, incidentally, that b g = 0 in the quenched approximation, because the purely gluonic observables (such as g 2 ) are automatically improved at non-zero quark mass if they are at zero quark mass. In the case of the coecients b m, b A and b P only the tree-level result b (0) m = 1 2 ; (3:19) b (0) A = b(0) P = 1; (3:20) is currently available [8,23]. The corrections associated with these coecients may be non-negligible at the larger quark masses. 3.4 Renormalization conditions A complete specication of a mass-independent renormalization scheme requires that we x the nite parts of the renormalization constants Z g, Z m and Z by imposing an appropriate set of renormalization conditions. Dierent schemes are then related by transformations of the form g 2 R! g 2 RX g (g 2 R); m R! m R X m (g 2 R); R! R X (g 2 R) (3:21) (up to corrections of order a 2 ). For perturbation theory minimal subtraction is a technically attractive renormalization prescription. It is dened by the requirement that the expansion coecients Z (l) g, Z (l) m and Z (l) are polynomials in ln(a) with no constant term, to any order l 1 of perturbation theory. At the non-perturbative level mass-independent renormalization schemes are not as easy to dene. For such a scheme to be practically useful, the renormalization constants Z g, Z m and Z should be computable through numerical simulations. Since they refer to properties of the theory at zero quark mass, one is then confronted with the problem of simulating QCD with very light (or even massless) quarks. A scheme where this diculty is bypassed has been described in ref. [2]. The proposition is to impose all renormalization conditions on a set of correlation functions derived from the QCD Schrodinger functional. In this framework an infrared cuto is provided by the nite extent L of the lattice (given in physical units) and one may then safely set the quark mass to zero without running 15

17 into singular quark propagators. The philosophy which goes along with this approach has been explained in ref. [2] and a possible choice of renormalization conditions is described there. 4. Schrodinger functional Most of the details given in this section have previously appeared in the literature [13{15,17]. Our aim is not to motivate the use of the Schrodinger functional or to explain its basic properties. For this the reader should consult the papers quoted above and also refs. [20,21], where an alternative approach (using the temporal gauge and spectral boundary conditions for the quark and anti-quark elds) is discussed. Instead we would like to briey recall the relevant denitions and to set up the notations that will be used in the rest of this paper and in refs. [23{25]. We rst introduce the Schrodinger functional without O(a) improvement and derive the required correction terms in sect Lattice geometry and elds In general the formulation of the theory is as in sect. 2 except that the lattice is now taken to be of nite extent in all directions. The possible values of the time coordinate x 0 of a lattice point x are then x 0 = 0; a; 2a; : : :; T. At xed times the lattice is thought to be wrapped on a torus of size L 3. In other words, all elds are assumed to be periodic functions of the space coordinates with period L. At the boundaries x 0 = 0 and x 0 = T we impose Dirichlet boundary conditions. In the case of the gauge eld the requirement is that U(x; k)j x0 =0 = W (x; k); U(x; k)j x 0 =T = W 0 (x; k); (4:1) where W (x; k) and W 0 (x; k) are some externally prescribed elds. More precisely, we choose a smooth continuum gauge eld C k (x) and set Z 1 W (x; k) = P exp a dt C k (x + a^k ta^k) : (4:2) 0 Similarly, W 0 (x; k) is given by another eld C 0 k (x). In eq. (4.2) the symbol P implies a path-ordered exponential such that the elds at the larger values of 16

18 t come rst. Note that the temporal gauge eld variables U(x; 0) (which are dened for 0 x 0 < T ) remain unconstrained. The dynamical degrees of freedom of the quark and anti-quark elds (x) and (x) reside on the lattice sites x with 0 < x 0 < T. At the boundaries only half of the Dirac components are dened and these are xed to some prescribed values ; : : :; 0. Explicitly, if we introduce the projectors P = 1 2 (1 0), the boundary conditions on the quark eld are P + (x)j x0 =0 = (x); P (x)j x0 =T = 0 (x); (4:3) while for the anti-quark eld we require that (x)p j x0 =0 = (x); (x)p + j x0 =T = 0 (x): (4:4) For consistency the boundary values must be such that the complementary components P ; : : :; 0 P vanish. 4.2 Action In the interior of the lattice the action density is given by the same expressions as on the innite lattice considered previously. We however need to specify the precise form of the action in the vicinity of the boundaries x 0 = 0 and x 0 = T. In particular, for the Wilson plaquette action we now write S G [U] = 1 g 2 0 X p w(p) tr f1 U(p)g; (4:5) where the sum runs over all oriented plaquettes p whose corners have time coordinates x 0 in the range 0 x 0 T. The weight w(p) is equal to 1 for all p except for the spatial plaquettes at x 0 = 0 and x 0 = T, which are given the weight 1 2. To be able to write the quark action in an elegant form it is useful to extend the elds to all times x 0 by \padding" with zeros. In the case of the quark eld this amounts to setting (x) = 0 if x 0 < 0 or x 0 > T ; (4:6) and P (x)j x0 =0 = P + (x)j x0 =T = 0: (4:7) 17

19 The anti-quark eld is extended similarly and the so far undened link variables are set to 1. With this notational convention the Wilson-Dirac operator (2.3) is well-dened at all times and the quark action is again given by eq. (2.4). An important detail we should mention at this point is that an unconventional phase factor has been included in the denition (A.13){(A.17) of the covariant lattice derivatives r and r. This factor is equivalent to imposing the generalized periodic boundary conditions (x + L^k) = e i k (x); (x + L^k) = (x)e i k ; (4:8) as one easily proves by performing an abelian gauge transformation. The present formulation, where the elds are strictly periodic and appears in the dierence operators, is technically simpler. In any case, the angles k parametrize a family of admissable boundary conditions and give us further opportunities to probe the quark dynamics. 4.3 Functional integral and correlation functions The Schrodinger functional Z[C 0 ; 0 ; 0 ; C; ; ] = Z D[U]D[ ]D[ ] e S[U; ; ] (4:9) involves an integration over all elds with the specied boundary values. Note that the integration measure does not depend on the boundary values of the elds. All dependence on the latter derives from the action. The expectation value of any product O of elds is given by hoi = 1 Z Z D[U]D[ ]D[ ] O e S[U; ; ] : (4:10) 0 = 0 ===0 Apart from the gauge eld and the quark and anti-quark elds integrated over, O may involve the \boundary elds" (x) = 0 (x) = (x) ; (x) = (x) ; (4:11) 0 (x) ; 0 (x) = 0 (x) : (4:12) These are perfectly meaningful in eq. (4.10) (the derivatives act on the Boltzmann factor) and have the eect of inserting certain combinations of (x) 18

20 and (x) close to the boundaries of the lattice, together with the appropriate gauge eld variables to ensure gauge covariance (see appendix C for the precise denition of the variational derivatives). 5. O(a) improvement of the Schrodinger functional The reader may wish to skip this section on rst reading, because the discussion of the PCAC relation in sect. 6 is only marginally dependent on the results obtained here. For a more solid understanding of our approach it is however useful to know that the Schrodinger functional is well-behaved in the continuum limit and that the improvement programme extends to this quantity in a straightforward manner. A further motivation to include this section is that we would like to prepare the ground for the computation of the running coupling and the running quark mass along the lines described in ref. [2]. In the case of the pure gauge theory, the problem of the renormalization and O(a) improvement of the Schrodinger functional has already been addressed in subsects. 2.5 and 4.5 of ref. [15]. The argumentation presented there carries over literally to QCD with any number of quarks. We shall, therefore, restrict the discussion to those aspects that we believe are new or worthwhile to be reconsidered. 5.1 Continuum limit and scope of improvement In the following the boundary values C, C 0, ; : : :; 0 are taken to be linear combinations of a nite number of plane waves with coecients and momenta that are kept xed as the lattice spacing is sent to zero. Since we assume periodic boundary conditions in the space directions, the momenta must be integer multiples of 2=L. As for the correlation functions we restrict attention to the expectation values of products of local composite elds (x) and the Fourier components of the boundary quark elds (y); : : :; 0 (z). We assume that the elds (x) are inserted at non-zero (physical) distances from the boundaries and from each other. The renormalization of the Schrodinger functional is achieved by expressing the bare parameters through the renormalized parameters and by scaling the boundary values of the quark and anti-quark eld with a renormalization constant Z [15,17]. For xed renormalized boundary values, as specied 19

21 above, the continuum limit can then be taken and one ends up with a welldened functional of the boundary values of the gauge and quark elds. By dierentiating with respect to the latter, the continuum limit of the correlation functions of the Fourier components of the boundary elds (y); : : :; 0 (z) is obtained at the same time. Operator insertions in the interior of the space-time volume require additional renormalizations as in innite volume. It should be noted, however, that the operator insertions and the boundary elds are treated quite dierently. In the rst case the insertions are made at positions that are at non-zero distances from each other and from the boundaries. A discussion of short distance singularities is thus avoided, and the renormalizations needed are just those that are already required in on-shell matrix elements of the operators. The correlation functions of the boundary elds, on the other hand, are obtained in momentum space by dierentiation of the Schrodinger functional. Arbitrary (nite) products of the Fourier components of these elds can occur. Short distance singularities associated with such products are automatically taken care of on the level of the Schrodinger functional, where they would show up in the form of divergent local boundary terms, constructed from the boundary values of the elds. Our aim in this section is to study the cuto dependence of the correlation functions of the type mentioned above and to determine the required O(a) counterterms. The improvement of the elds (x) is as in innite volume and will not be considered again. We are then left with the correlation functions of the boundary quark elds. As in the case of the renormalization of the theory, it is simpler to discuss the improvement of the Schrodinger functional itself. Once the latter is improved the correlation functions will be improved, too. In particular, no further subtractions will be needed, and the O(a) counterterms (y); : : :; 0 (z) are all equal to zero. 5.2 Boundary terms and eective theory The approach of the Schrodinger functional to the continuum limit can be described by a local eective theory. Compared to our discussion in sect. 2 of the innite volume theory the principal dierence is that we now need to include further terms in the eective action to account for boundary eects. The general form of the correction terms in eq. (2.5) then is Z Z S k = d 4 x L k (x) + lim d 3 x B k (x)j!0 x0 = + B 0 k(x)j x0 =T ; (5:1) where B k (x) and B k 0 (x) are linear combinations of local composite elds of 20

22 dimension 3 + k. A technical remark we should make here is that the limit! 0 is non-trivial in general and requires that the coecients in the linear combinations are scaled in a particular way [13,14]. This complication is not very important in the present context, because we are only interested in the general form of the boundary terms, which is determined by symmetry considerations. The relevant symmetries are all internal symmetries of the lattice theory and the discrete space rotations and reections. Moreover B k (x) and B k 0 (x) are related by a time reection so that only one of them needs to be discussed. In subsect. 2.3 we have shown that under certain conditions the eld equations may be used to reduce the number of terms contributing to the eective action. The same arguments may be applied here, rst to simplify the eective lagrangian L 1 (x) and then also the boundary terms B 1 (x) and B 1 0 (x). The limit! 0 is helpful at this point, because the applicability of the eld equations to reduce the number of terms is made obvious. As for the volume term, L 1 (x), we conclude that it can be taken to be the same as in the innite volume theory. For the improved lattice action we thus write S impr [U; ; ] = S[U; ; ] + S v [U; ; ] + S G;b [U] + S F;b [U; ; ]; (5:2) TX a S v [U; ; ] = a 5 x 0 =a X x c sw (x) i 4 b F (x) (x); (5:3) where the notation is as in subsect. 2.4 (the index \v" indicates a volume counterterm, while the boundary counterterms are labeled by \b"). In the following we deduce the form of the boundary counterterms. 21

23 Table 1. Improvement coecients c (k;l) s;t N c (1;0) s c (1;0) t c (1;1) t c (2;0) t 2 0:166(1) [15] 0:0543(5) [15] 0: (1)[18] 0:0115(5) [19] 3 0:08900(5) [16] 0: (1)[18] 5.3 Boundary counterterms independent of the quark elds In the pure gauge theory any gauge invariant local composite eld has dimension greater or equal to 4. The only elds that can contribute to B 1 and B 1 0 are hence given by O 9 = tr ff kl F kl g ; (5:4) O 10 = tr ff 0k F 0k g : (5:5) For the associated O(a) counterterm we may take [15] S G;b [U] = 1 X 2g0 2 (c s 1) tr f1 U(p s )g p s + 1 X g0 2 (c t 1) tr f1 U(p t )g : (5:6) p t The sums here run over all oriented plaquettes p s and p t at the boundaries that are space-like (p s ) or time-like (p t ). The notation is otherwise as in sect. 2. In perturbation theory we have c s;t = 1 + c (1) s;t g2 0 + c(2) s;t g4 0 + : : :; (5:7) c (k) s;t = kx l=0 c (k;l) s;t N l f : (5:8) The known expansion coecients for N = 2 and N = 3 are listed in table 1 (the numbers in square brackets point to the references from which the coecients have been taken). 22

24 5.4 Boundary counterterms depending on the quark elds The presence of the quark elds allows one to construct many more composite elds of dimension 4 with the required symmetries. A basis of such elds is given by O 11 = P + D 0 + D 0 P ; (5:9) O 12 = P D 0 + D 0 P + ; (5:10) O 13 = P + k D k D k k P ; (5:11) O 14 = P k D k D k k P + ; (5:12) O 15 = m : (5:13) The formal eld equations imply two relations, O 11 + O 13 + O 15 = 0; (5:14) O 12 O 14 O 15 = 0; (5:15) so that two elds can be eliminated. A particularly simple form of the associated boundary counterterms to the lattice action is obtained if we choose O 11 ; O 14 ; O 15 at x 0 = 0; (5:16) O 12 ; O 13 ; O 15 at x 0 = T; (5:17) as the basis of elds. Note that we are free to choose a dierent basis at the two boundaries of the lattice. The counterterm corresponding to O 15 is of a special kind, similar to the volume counterterms associated with O 3 and O 5 (cf. sect. 2). To see this we consider the term at x 0 = 0 and choose the lattice eld, representing O 15, to be given by m q (y)u(x; 0) 1 P + (x) + (x)p U(x; 0) (y) y=x+a^0 : (5:18) Next we note that the terms in the Wilson quark action that are linear in the boundary values and are precisely proportional to the above expression. 23

25 The boundary counterterm can thus be compensated by rescaling and by a factor of the form 1 + b(g0)am 2 q. There is also a term in the action depending quadratically on the boundary values, but since it is already of O(a) the rescaling of this term produces a correction of O(a 2 ) which is considered negligible in the present discussion. Since one anyway has to renormalize the boundary values of the quark and anti-quark elds, the O(a) counterterms corresponding to O 15 are hence not included in the improved action. If a mass-independent renormalization scheme is employed, the counterterm instead appears in the denition R (x) = Z (~g 2 0; a)(1 + b am q )(x) (5:19) of the renormalized boundary eld R (x) (the other boundary elds are renormalized similarly). We are thus left with altogether four boundary counterterms, two at time x 0 = 0 and two at x 0 = T. Their coecients must be such that the time reversal invariance of the theory is preserved. A possible choice of the counterterms then is S F;b [U; ; ] = a 4 X x n(~c s 1) b Os (x) + b Os 0 (x) + (~c t 1) b Ot (x) b Ot 0 (x) o; (5:20) where bo s (x) = 1 2 (x) k(r k + r k )(x); (5:21) bo s 0 (x) = (x) k (r k + r k ) 0 (x); (5:22) bo t (x) = (y)p + r 0 (y) + (y)r 0P (y) y=(a;x) ; (5:23) bo t 0 (x) = (y)p r 0 (y) + (y)r 0 P + (y) y=(t a;x) : (5:24) Note that one does not refer to any \undened" components of the quark and anti-quark elds in these expressions. The projectors P and the time derivatives are always such that this is avoided. 24

26 As will be shown in ref. [23] the Schrodinger functional in the free quark theory with Wilson action is already O(a) improved. The perturbation expansion of the improvement coecients ~c s and ~c t is hence of the form ~c s;t = 1 + ~c (1) s;t g2 0 + ~c(2) s;t g4 0 + : : : (5:25) It may seem a bit articial to write the coecients in eq. (5.20) in the way we did. The notation was chosen to emphasize the analogy with the pure gauge counterterm (5.6). Moreover the terms appearing in eq. (5.20) combine with the terms in the Wilson quark action. At x 0 = 0, for example, we have a b Ot (x) = (x + a^0) (x + a^0) (x + a^0)u(x; 0) 1 P + (x) (x)p U(x; 0) (x + a^0): (5:26) This counterterm thus contributes to the mass term at x 0 = a and changes the weight of the time-like hopping terms at the boundary from 1 to ~c t. 25

27 6. Chiral symmetry restoration We now proceed to study the axial current conservation on the lattice. The coecients c sw, c A, etc. are initially assumed to have their proper values so that the theory is on-shell O(a) improved. Chiral symmetry in lattice QCD with Wilson quarks has been extensively discussed in the literature (an incomplete list of references is [26{29]). Here we show that O(a) improvement leads to a signicant reduction of the cuto eects in the PCAC relation. This may conversely be taken as an \improvement condition" to x c sw and c A. 6.1 PCAC relation In a mass-independent renormalization scheme, the renormalized improved axial current and density are given by (A R ) a (x) = Z A (~g 2 0; a)(1 + b A am q ) A a (x) + A a (x) ; (6:1) (P R ) a (x) = Z P (~g 2 0; a)(1 + b P am q )P a (x) (6:2) (cf. subsect. 2.5). The renormalization constants Z A and Z P should be xed by imposing appropriate renormalization conditions. In what follows we do not need to know exactly how this is done. We shall however take it for granted that the normalization mass is independent of the lattice size (this is the normal situation, but the remark is made here, because in the SF scheme discussed in ref. [2] one sets = 1=L). We now rst consider the theory in innite volume. Since the PCAC relation (1.1) holds in the continuum limit, and since the improvement has been fully implemented, we expect that 1 2 (@ )(A R ) a (x) O = 2m R (PR ) a (x) O + O(a 2 ); (6:3) for any product O of renormalized improved elds located at non-zero distances from x and from each other. As already indicated by the notation, the proportionality constant m R is a renormalized quark mass which can be shown to be of the form (3.15) up to corrections of order a 2. There is one aspect of eq. (6.3) which may appear to be trivial but which will be most important in what follows. The property we wish to emphasize is that the relation holds for any product O of elds localized in a region not 26

28 containing x. Of course this is just a reection of the fact that the PCAC relation becomes an operator identity when the theory is formulated in Minkowski space. Dierent choices of O then correspond to considering dierent matrix elements of the relation. It may nevertheless be useful to briey recall the standard proof of eq. (6.3) in the euclidean framework. First note that the lattice correlation functions on both sides of the equation converge to the continuum limit with a rate proportional to a 2. It is hence sucient to establish eq. (6.3) in the continuum theory. To this end one performs a local innitesimal chiral transformation on the quark elds integrated over in the functional integral. We can choose the transformation to be trivial outside a small neighbourhood R of the point x. The variation S 0 of the action is proportional to Z R d 4 y! a (A R ) a (y) 2m R (P R ) a (y) ; (6:4) where! a (y) is the (position dependent) transformation parameter. The product O is localized outside the region R and is hence not aected by the transformation. The invariance of the quark integration measure then implies hs 0 Oi = 0; (6:5) which reduces to eq. (6.3) after passing to the local limit. Exactly the same argumentation may be applied if the theory is set up in a nite volume with Schrodinger functional boundary conditions. We are thus led to conclude that eq. (6.3) must be valid in this situation too, with the same value of m R, provided the point x is at a non-zero distance from the space-time boundaries. At rst sight it may not be totally obvious that the proportionality constant m R is indeed independent of the lattice sizes T and L (up to corrections of order a 2 ). A second derivation of this important fact is therefore given in appendix D. We conclude this subsection by noting that eq. (6.3) remains valid if O is taken to be a product of unimproved elds or if the O(a) boundary counterterms S G;b and S F;b are dropped. In other words, to ensure that the lattice corrections in eq. (6.3) are of order a 2, it is enough to include the Sheikholeslami-Wohlert term in the action and to insert the renormalized improved axial current and density. The reason for this is best explained by considering a simple example. Let us assume that O is equal to some renormalized improved eld R (y) and 27